= 3^rd^ order Runge-Kutta scheme =

For the dicretization in time a 3^rd^ order low-storage Runge-Kutta scheme with 3 stages recommended by Williamson (1979) is used.
Generally an N-stage Runge-Kutta scheme discretizes an ordinary differential equation of the form
{{{
#!Latex
\[  \frac{d \psi}{d t} & = & f(t,\psi) \]
}}}
as follows ( Baldauf, 2008 ):
{{{
#!Latex
\[ \psi^{(0)} = \psi^{n}, \]
\[ k^{i} = f(t^{n} + \Delta t\,\alpha_{i},\,\psi^{i-1}), \]
\[ \psi^{i} = \psi^{n} + \Delta t\,\sum^{i}_{j=1}\,\beta_{i+1,j}\,k^{j}, \quad \textnormal{mit} \quad i \in [1,2,...,N] \]
\[ \psi^{n+1} = \psi^{N}. \]
}}}
The coefficients can written in a Butcher-Tableau in the following way:
|| α,,1,, || β,,1,1,, || 0 || ... || || ||
|| α,,2,, || β,,2,1,, || β,,2,2,, || 0 || ... || ||
|| ...    ||  ... || ||  ||  || ||
|| α,,N,, || β,,N,1,, || β,,N,2,, || ... || β,,N,N-1,, || 0 || 
||        || β,,N+1,1,, || β,,N+1,2,, || ... || β,,N+1,N-1,, ||  β,,N+1,N,, || 

The appendant coefficients for the used Runge-Kutta scheme reads:
||   0  ||   0   ||   0   ||   0  ||
||  1/3 ||    1/3 ||    0  ||  0   ||
|| 1/2  ||  -3/16 || 15/16 ||  0  ||
||      ||  1/6  ||  3/10 || 8/15 ||

For the implementation it is advantageous to compute ψ^N^ from the intermediate solutions ψ^1^ and ψ^2^ and combine the local tendencies in one array after the second substep to save storage (therefor low-storage scheme) as follows: 
{{{
#!Latex
\[ \hat\psi_{1} & = & \psi_{n} + \frac{1}{3} \Delta t f\left(\psi_{n}\right) \]
\[ \hat\psi_{2} & = & \hat\psi_{1} + \frac{1}{48} \Delta t \left( 45 f\left(\hat\psi_1\right) - 25 f\left(\psi_{n}\right) \right) \]
\[ f\left(\hat\psi_{1}\right) & = &-153 f\left(\hat\psi_{1}\right) + 85 f\left(\psi_{n}\right) \]
\[ \hat\psi_{3} & = & \left( \psi_{n+1} \right) = \hat\psi_{2} + \frac{1}{240} \Delta t \left( 128 f\left(\hat\psi_2\right) + 15 f \left(\hat\psi_{1}\right) \right) \]
}}}

For reasons of clarity the time integration for several schemes (further schemes are: Leap-frog, Euler and 2^nd^ order Runge-Kutta scheme) is implemented as following (here for example the u-component of velocity):

{{{
#!Latex
\begin{split}
 \textnormal{u}\_\textnormal{p}\left(k,j,i\right) = \left(1.0 - \textnormal{tsc}\left(1\right) \right) * \textnormal{u}\_\textnormal{m}\left(k,j,i\right) + \textnormal{tsc}\left(1\right) * \textnormal{u}\left(k,j,i\right) + \textnormal{dt}\_\textnormal{3d}* \left( \\ 
                             \textnormal{tsc}\left(2\right) * \textnormal{tend}\left(k,j,i\right) + \textnormal{tsc}\left(3\right) * \textnormal{tu}\_\textnormal{m}\left(k,j,i\right)  \\
                             + \textnormal{tsc}\left(4\right) * \left(\textnormal{p}\left(k,j,i)\right) - \textnormal{p}\left(k,j,i-1\right) \right) * ddx \ \ ) \\
                             - \textnormal{tsc}\left(5\right) * \textnormal{rdf}\left(k\right) * \left(\textnormal{u}\left(k,j,i) - \textnormal{ug} \right) 
\end{split}
}}}

and steered by the array tsc(1:5)

|| tsc(1) || tsc(2) || tsc(3) || tsc(4) || tsc(5) ||
|| 1 || 1/3 || 0 || 0 || 0 || 1^st^ substep
|| 1 || 15/16 || -25/48 || 0 || 0 || 2^nd^ substep
|| 1 || 8/15 || 1/15 || 0 || 1 || 3^rd^ substep



Sorry for incompleteness. A further description will follow in the next days.